Introduction

� A power-flow study (load-flow study) is an analysis of the voltages, currents, and power flows in a power system under steady-state conditions.

� In such a study, we make an assumption about either

a voltage at a bus or the power being supplied to the bus for each bus in the power system and then determine the magnitude and phase angles of the bus voltages, line currents, etc.

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Basic techniques for power-flow studies

The simplest way to perform power-flow calculations is by iteration:

1. Create a bus admittance matrix Ybus for the power system;

2. Make an initial estimate for the voltages at each bus in the system;

3. Update the voltage estimate for each bus (one at a time).

4. Repeat this process to make the voltages at each bus approaching the correct answers closer and closer…

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Basic techniques for power-flow studies

Each bus in a power system can be classified to one of three types: 1) Load bus (PQ bus) – a bus at which the real and reactive

power are specified, and for which the bus voltage will be calculated. Real and reactive powers supplied to a power system are defined to be positive, while the powers consumed from the system are defined to be negative. All busses having no generators are load busses.

2) Generator bus (PV bus) – a bus at which the magnitude of the voltage is kept constant. we can control and specify the magnitude of the bus voltage and real power supplied.

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Basic techniques for power-flow studies

3) Slack bus (swing bus) – a special generator bus serving as

the reference bus for the power system. Its voltage is assumed to be fixed in both magnitude and phase (for instance, 1 0° pu). The real and reactive powers are uncontrolled: the bus supplies whatever real or reactive power is necessary to make the power flows in the system balance.

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Basic techniques for power-flow studies

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Basic techniques for power-flow studies

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Power Flow Equation

Consider a typical bus of the power system:

Applying KCL to this bus results in

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Power Flow Equation

The real and reactive power at bus i is :

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PPower-flow analysis

This method is known as the Gauss-Siedel iterative method. Its basic procedure is:

1) Calculate the bus admittance matrix Ybus including the admittances of all transmission lines, transformers, etc., between busses but exclude the admittances of the loads or generators themselves.

2) Select a slack bus: one of the busses in the power system, whose voltage will arbitrarily be assumed as 1.0 0°.

3) Select initial estimates for all bus voltages: usually, the voltage at every load bus assumed as 1.0 0° (flat start) lead to good convergence.

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PPower-flow analysis

4) Write voltage equations for every other bus in the system. The generic form is

5) Calculate an updated estimate of the voltage at each load bus in succession using the above equation except for the slack bus.

6) Compare the differences between the old and new voltage estimates: if the differences are less than some specified tolerance for all busses, stop. Otherwise, repeat step 5.

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PPower-flow analysis (Example)

Example : In a 2-bus power system, a generator attached to bus 1 and loads attached to bus 2. The series impedance of a single transmission line connecting them is 0.1+j0.5 pu. The shunt admittance of the line may be neglected. Assume that bus 1 is the slack bus and that it has a voltage V1 = 1.0 0° pu. Real and reactive powers supplied to the loads from the system at bus 2 are P2 =0.3 pu, Q2 =0.2 pu (powers supplied to the system at each busses is negative of the above values). Determine voltages at each bus for the specified load conditions.

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PPower-flow analysis (Example)

Solution : Step 1. We start from calculating the bus admittance matrix Ybus. The series admittance of line 1 is: Applying similar calculations to other terms, we complete the admittance matrix as:

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PPower-flow analysis (Example)

Solution : Step 2. Next, we select bus 1 as the slack bus since it is the only

bus in the system connected to the generator. The voltage at bus 1 will be assumed 1.0 0°.

Step 3. We select initial estimates for all bus voltages. Making a

flat start, the initial voltage estimates at every bus are 1.0 0°.

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PPower-flow analysis (Example)

Solution : Step 4. Next, we write voltage equations for every other bus in

the system. For bus 2:

Since the real and reactive powers supplied to the system at bus 2 are P2 = -0.3 pu and Q2 = -0.2 pu and since Ybus and V1 are known, we may reduce the last equation:

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PPower-flow analysis (Example)

Step 5. Next, we calculate an updated estimate of the voltages at each load bus in succession. In this problem we only need to calculate updated voltages for bus 2, since the voltage at the slack bus (bus 1) is assumed constant. We repeat this calculation until the voltage converges to a constant value. The initial estimate for the voltage is V2,0 = 1 0°. The next estimate for the voltage is :

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PPower-flow analysis (Example)

This new estimate for V2 substituted back to the equation will produce the second estimate: The third iteration will be The fourth iteration will be

e

e

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PPower-flow analysis (Example)

The fifth iteration will be : We observe that the magnitude of the voltage is barely changing and may

conclude that this value is close to the correct answer and, therefore, stop the iterations.

This power system converged to the answer in five iterations. The voltages at each bus in the power system are:

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PPower-flow analysis (Example)

If we compare the differences between the old and new voltage estimates:

This power system converged to the answer for tolerance ε < 0.001

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PPower-flow analysis (Example)

The current flowing from bus 1 to bus 2 is :

The power supplied by the transmission line to bus 2 is :

P2 Q2

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AAdding generator busses

At a generator bus, the real power Pi and the magnitude of the bus voltage |Vi| are specified. Since the reactive power for that bus is usually unknown, we need to estimate it before getting updated voltage estimates. The value of reactive power at the generator bus can be estimated by solving the following equation for Qi:

Bringing the case k = i, into summation, we obtain:

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AAdding generator busses

� Once the reactive power at the bus is estimated, we can

update the bus voltage at a generator bus using Pi and Qi as we would at a load bus.

� However, the magnitude of the generator bus voltage is also forced to remain constant. Therefore, we must multiply the new voltage estimate by the ratio of magnitudes of old to new estimates.

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AAdding generator busses

Therefore, the steps required to update the voltage at a generator bus are:

1. Estimate the reactive power Qi ; 2. Update the estimated voltage at the bus as if the bus

was a load bus; 3. Force the magnitude of the estimated voltage to be

constant by multiplying the new voltage estimate by the ratio of the magnitude of the original estimate to the magnitude of the new estimate. This has the effect of updating the voltage phase estimate without changing the voltage amplitude.

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Example

Example : The Ybus matrix of a three-bus power system is given as follow. Compute bus voltages for tolerance ε < 0.05.

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Example

Solution : First of all, we consider bus 1 as the slack bus and select initial estimates for all bus voltages.

Bus 2 is PV bus. Since in PV buses Pi and |Vi| are constant, the initial estimate for |V2| must be 1.05 p.u, which is the fixed value. The initial estimate for δ2 is 0o , but it would be changed in each iteration.

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Example

First Iteration : (Bus 2 , PV bus)

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Example

First Iteration : (Bus 2 , PV bus)

Unchanged

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Example

First Iteration : (Bus 3 , Load bus)

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Example

First Iteration :

The stopping criterion is not satisfied. We have to continue the calculations for the 2nd iteration.

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Example

Second Iteration : (Bus 2 , PV bus)

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Example

Second Iteration : (Bus 2 , PV bus)

Unchanged

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Example

Second Iteration : (Bus 3 , Load bus)

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Example

Second Iteration : 33